Problem 37 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

In a group of boys, two boys are brothers and in this group 6 more boys are there. In how many ways they can sit if the brothers are not to sit along with each other (A) 4820 (B) 1410 (C) 2830 (D) None of these

Step-by-Step Solution

Answer

The answer is (D) None of these.

Step 1: Understanding the Problem

We have a total of 8 boys in a group, of which two are brothers. They need to be seated such that these brothers do not sit next to each other. Our task is to find the number of seating arrangements that satisfy this condition.

Step 2: Calculate Total Arrangements Without Restriction

First, we find the number of ways to arrange all 8 boys without any restrictions. This is simply 8 factorial, given by: \[8! = 40320\]

Step 3: Calculate Arrangements with Brothers Together

Treat the two brothers as a single unit or a 'block'. This reduces the number of units to arrange from 8 to 7. The brothers within the block can switch places, so we have:\[7! \times 2! = 5040 \times 2 = 10080\] ways to arrange the group with brothers together.

Step 4: Subtract Arrangements with Brothers Together

To find arrangements where the brothers do not sit together, subtract the number where they sit together from the total:\[8! - (7! \times 2!) = 40320 - 10080 = 30240\]

Step 5: Short Listing Correct Option

Since none of these options match with the calculated value of 30,240, the correct answer must be (D) None of these.

Key Concepts

Factorial

The concept of a factorial, denoted by an exclamation mark (!), is a fundamental building block in permutations and combinations. It represents the product of all positive integers up to a given number. For instance, the factorial of a number \( n \) is written as \( n! \) and calculated as follows:

• \( 0! = 1 \)
• \( 1! = 1 \)
• \( 2! = 2 \times 1 = 2 \)
• \( 3! = 3 \times 2 \times 1 = 6 \)
• \( n! = n \times (n-1) \times (n-2) \times \, ... \, \times 3 \times 2 \times 1 \)

The factorial is useful when calculating the total number of arrangements of a set, which involves multiplying all descending whole numbers. For example, \( 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40,320 \). This tells us there are 40,320 ways to arrange 8 distinct objects.

Seating Arrangement

The concept of seating arrangement involves finding the number of ways to organize people, objects, or entities in a line or around a table. This is a common problem in permutation, where the order matters.
In the given exercise, we have 8 boys, including two brothers. Just looking for how we can arrange everyone without caring about the seating restriction involves:

• Calculating \( 8! \), which gives us the total arrangements of 8 people.

However, an additional layer of complexity is added when we introduce restrictions, such as ensuring two brothers do not sit together. Such restrictions often require:

• Considering certain entities as a single unit, also known as a 'block',.
• Counting arrangements as if they were a single item.
• Adjusting the total arrangement using combinations and permutation methods.

This strategy gives a clear picture of how different arrangements can be formed under various conditions.

Exclusion Principle

The exclusion principle, often crucial in solving problems where there are restrictions, involves calculating the total possible outcomes without applying restrictions and then subtracting the restricted cases.
In this problem, we initially calculate the total possible arrangements of the boys (without restriction) using \( 8! \). The restricted condition is that the two brothers should not sit together.

• By treating brothers as a unit, calculate arrangements where they sit together: \( 7! \times 2! \).
• Subtract these restricted arrangement outcomes from the total arrangements.

This final subtraction gives us the desired seating patterns where the specific restriction is met. It's like ensuring that the restricted event does not occur in any of the outcomes. By excluding the undesirable possibilities, we narrow down to only the valid configurations.